Gorontalo Utara

Boolean Algebra Expression Laws, Rules, Theorems and Solved Examples

Published

3 years ago

18/08/2022

There are some set of logical expressions which we accept as true and upon which we can build a set of useful theorems. These sets of logical expressions are known as Axioms or postulates of Boolean Algebra. An axiom is nothing more than the definition of three basic logic operations (AND, OR and NOT). All axioms defined in boolean algebra are the results of an operation that is performed by a logical gate.

In contrast, in a list of some but not all of the same laws, there could have been Boolean laws that did not follow from those on the list, and moreover there would have been models of the listed laws that were not Boolean algebras. These generalized expressions are very important as they are used to simplify many Boolean Functions and expressions. Minimizing the boolean function is useful in eliminating variables and Gate Level Minimization. The theorem states that the complement of the OR operation between two or more variables is equivalent to the AND operation of their complements.

\(B\) is
unique up to \(A\)-isomorphism, and is called the completion of
\(A\). If \(f\) is a homomorphism from a BA \(A\) into a complete BA
\(B\), and if \(A\) is a subalgebra https://1investing.in/ of \(C\), then \(f\) can be
extended to a homomorphism of \(C\) into \(B\). Another general algebraic notion
which applies to Boolean algebras is the notion of a free
algebra.

Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitrary axiomatic system will not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything. Informally, this infinite set of axioms states that there are infinitely many different items. However, the concept of an infinite set cannot be defined within the system — let alone the cardinality of such as set. In electrical and electronic circuits, Boolean algebra is used to simplify and analyze the logical or digital circuits. The other theorems in Boolean algebra are complementary theorem, duality theorem, transposition theorem, redundancy theorem and so on. All these theorems are used to simplify the given Boolean expression.

Furthermore, Boolean algebras can then be defined as the models of these axioms as treated in § Boolean algebras. The equivalent logical operators to these operations are given below. Boolean algebra is a type of algebra that is created by operating the binary system. In the year 1854, George Boole, an English mathematician, proposed this algebra. This is a variant of Aristotle’s propositional logic that uses the symbols 0 and 1, or True and False.

Mathematical methods developed to some degree of sophistication in ancient Egypt, Babylon, India, and China, apparently without employing the axiomatic method. Embark on a transformative journey towards GATE success by choosing Data Science & AI as your second paper choice with our specialized course. If you find yourself lost in the vast landscape of the GATE syllabus, our program is the compass you need. Boolean Algebra also called Logical Algebra is a branch of mathematics that deals with Boolean Varaibles such as, 0 and 1. Of course, it is possible to code more than two symbols in any given medium.

The shading indicates the value of the operation for each combination of regions, with dark denoting 1 and light 0 (some authors use the opposite convention).
However, we could put a circle for x in those boxes, in which case each would denote a function of one argument, x, which returns the same value independently of x, called a constant function.
This result depends on the Boolean prime ideal theorem, a choice principle slightly weaker than the axiom of choice.
At times, it is not even clear which collection of axioms a proof appeals to.

The system has at least two different models – one is the natural numbers (isomorphic to any other countably infinite set), and another is the real numbers (isomorphic to any other set with the cardinality of the continuum). In fact, it has an infinite number of models, one for each cardinality of an infinite set. However, axiomatic definition of boolean algebra the property distinguishing these models is their cardinality — a property which cannot be defined within the system. A model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system.

Claude Shannon formally proved such behavior was logically equivalent to Boolean algebra in his 1937 master’s thesis, A Symbolic Analysis of Relay and Switching Circuits. The triangle denotes the operation that simply copies the input to the output; the small circle on the output denotes the actual inversion complementing the input. The convention of putting such a circle on any port means that the signal passing through this port is complemented on the way through, whether it is an input or output port. When values and operations can be paired up in a way that leaves everything important unchanged when all pairs are switched simultaneously, the members of each pair are called dual to each other. The duality principle, also called De Morgan duality, asserts that Boolean algebra is unchanged when all dual pairs are interchanged. Writing down further laws of Boolean algebra cannot give rise to any new consequences of these axioms, nor can it rule out any model of them.

Commutative Law

Boolean algebra is the category of algebra in which the variable’s values are the truth values, true and false, ordinarily denoted 1 and 0 respectively. It is used to analyze and simplify digital circuits or digital gates. It has been fundamental in the development of digital electronics and is provided for in all modern programming languages. The term “Boolean algebra” honors George Boole (1815–1864), a self-educated English mathematician. Boole’s formulation differs from that described above in some important respects.

Complementation Laws

This describes the scenario where the undefined terms of a first axiom system are provided definitions from a second, such that the axioms of the first are theorems of the second. The second law states that the complement of the sum of variables is equal to the product of their individual complements of a variable. The first law states that the complement of the product of the variables is equal to the sum of their individual complements of a variable. This means that if you want to find the complement of the OR operation of two or more variables, you can take the complement of each variable individually and then use the AND operation between their complements. This means that if you want to find the complement of the AND operation of two or more variables, you can take the complement of each variable individually and then use the OR operation between their complements. There are two basic theorems of great importance in Boolean Algebra, which are De Morgan’s First Laws, and De Morgan’s Second Laws.

Boolean algebra (structure)

Instead of showing that the Boolean laws are satisfied, we can instead postulate a set X, two binary operations on X, and one unary operation, and require that those operations satisfy the laws of Boolean algebra. The elements of X need not be bit vectors or subsets but can be anything at all. The closely related model of computation known as a Boolean circuit relates time complexity (of an algorithm) to circuit complexity. The theory of Boolean algebras was founded in 1847 by Boole, who considered it a form of ‘calculus’ adequate for the study of logic.

Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff’s 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.

For example, group theory was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously, without reference to the transformation group origins of those studies. A truth table represents all the combinations of input values and outputs in a tabular manner. All the possibilities of the input and output are shown in it and hence the name truth table. In logic problems, truth tables are commonly used to represent various cases.

OR Laws

The theorem states that the complement of the AND operation between two or more variables is equivalent to the OR operation of their complements. It is used to simplify logical circuits that are the backbone of modern technology. The inverse of the boolean variable is called the complement of the variable. A function of the Boolean Algebra that is formed by the use of Boolean variables and Boolean operators is called the Boolean function.

Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed in this way.[14][15][16] Boolean algebra is not sufficient to capture logic formulas using quantifiers, like those from first order logic. The two important theorems which are extremely used in Boolean algebra are De Morgan’s First law and De Morgan’s second law.

Using an axiomatic proof (i.e. using only the basic axioms and theorems of Boolean algebra).However, no matter what I do, I can’t seem to get things to line up correctly. Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory. That is, up to isomorphism, abstract and concrete Boolean algebras are the same thing. This result depends on the Boolean prime ideal theorem, a choice principle slightly weaker than the axiom of choice. This strong relationship implies a weaker result strengthening the observation in the previous subsection to the following easy consequence of representability.

Gorontalo

Menolak Lupa: Tragedi 2 Januari 2025, Ketika Keadilan untuk Julia Belum Datang

Published

1 week ago

27/11/2025

Redaksi Barakati

Misteri Kematian Julia Shinta: 11 Bulan Tanpa Titik Terang

Gorontalo – Sebelas bulan telah berlalu sejak tragedi memilukan yang menimpa seorang gadis muda bernama Julia Shinta Sangala, warga Kabupaten Gorontalo Utara. Pada 2 Januari 2025, jasad Julia ditemukan oleh seorang penggembala sapi di area semak-semak sepi di Desa Ketapang, Kecamatan Gentuma Raya. Penemuan itu seharusnya menjadi awal dari proses pengungkapan kebenaran dan penegakan keadilan bagi keluarga korban.

Sehari setelah penemuan, keluarga Julia melapor secara resmi ke Polres Gorontalo Utara. Dengan harapan besar, mereka mempercayakan penegakan hukum kepada pihak berwenang agar pelaku dapat segera ditemukan. Namun, waktu berjalan begitu lama tanpa perubahan berarti dalam proses penyidikan.

Sebelas Bulan Dalam Penantian Keadilan

Kini, 11 bulan telah berlalu, dan kasus tersebut masih menyandang status “dalam tahap penyidikan”. Tidak ada perkembangan signifikan yang disampaikan kepada publik, sementara keluarga terus menunggu kepastian hukum yang tak kunjung tiba.

Dalam kurun waktu hampir satu tahun, belum ada satu pun tersangka yang diumumkan. Tidak ada kejelasan tentang arah penyelidikan maupun hasil forensik yang dapat membuka tabir misteri kematian Julia Shinta Sangala.

Bagi keluarga korban, setiap hari terasa seperti menanggung luka yang sama. Mereka bukan hanya kehilangan anak, tetapi juga menghadapi ujian panjang melawan sistem hukum yang dinilai lambat dan tidak berpihak. Keadilan yang menjadi hak dasar warga seolah menjauh dan menjadi sesuatu yang sulit dijangkau.

Potret Suram Penegakan Hukum

Kasus ini bukan sekadar catatan kriminal di Gorontalo Utara. Ia merefleksikan wajah penegakan hukum di tingkat lokal yang tengah diuji. Ketika kasus pembunuhan dengan bukti dan peristiwa jelas tak kunjung menemukan titik terang selama hampir setahun, muncul pertanyaan besar tentang efektivitas dan keseriusan aparat dalam mengusut tuntas kejahatan.

Kondisi seperti ini tidak hanya melukai hati keluarga korban, tetapi juga menimbulkan kekhawatiran di tengah masyarakat. Rakyat perlu diyakinkan bahwa hukum benar-benar bekerja tanpa pandang bulu dan tidak terhambat oleh kepentingan apa pun.

Penulis Fikran Mohzen

Seruan dan Harapan

Kasus kematian Julia Shinta Sangala adalah panggilan moral bagi semua pihak. Masyarakat, pemerhati hukum, dan organisasi sipil diharapkan ikut mengawasi jalannya penyidikan agar berjalan transparan dan akuntabel.

Polres Gorontalo Utara diminta untuk memberikan penjelasan terbuka kepada publik terkait progres kasus. Keluarga korban berhak mendapatkan kepastian, bukan sekadar janji lanjutan proses penyidikan tanpa hasil yang jelas.

Sebagai bangsa yang menjunjung keadilan, tragedi ini tidak boleh dilupakan. Nama Julia Shinta Sangala harus terus diingat, bukan hanya sebagai korban, tetapi juga sebagai simbol perjuangan akan kebenaran di tengah sistem hukum yang lamban.

Keadilan mungkin tertunda, tetapi perjuangan untuk memperjuangkannya tidak boleh berhenti.

Keadilan untuk Julia.

Penulis
(Fikran Mohzen)

Gorontalo

Alarm Bahaya! PLTU Anggrek Diduga Buang Limbah Berbahaya ke Udara

Published

2 weeks ago

21/11/2025

Redaksi Barakati

Aktivis Provinsi Gorontalo, Isjayanto H. Doda.

Gorontalo – Aktivis Provinsi Gorontalo, Isjayanto H. Doda, memberikan peringatan keras kepada Pemerintah Daerah Kabupaten Gorontalo Utara (Gorut) agar segera mengambil langkah tegas terkait masalah Electrostatic Precipitator (ESP) di PLTU Anggrek yang diduga tidak berfungsi secara optimal.

Menurut penjelasan Isjayanto, limbah Fly Ash dan Bottom Ash (FABA) hasil pembakaran batu bara di PLTU tersebut diketahui mengandung zat logam berat yang membahayakan kesehatan manusia jika terpapar dalam jangka panjang.

“Jika sistem ESP PLTU Anggrek saat ini tidak berjalan baik, maka polusi udara dari FABA bisa dengan mudah terhirup oleh masyarakat sekitar. Ini sangat berbahaya jika partikel tersebut masuk ke paru-paru,” tegas Isjayanto dalam pernyataannya.

Ia menyoroti adanya ancaman serius yang mengintai masyarakat apabila kondisi ini dibiarkan berlarut-larut tanpa penanganan. Menghirup FABA, lanjutnya, dapat menimbulkan gangguan pernapasan serta dampak kesehatan jangka panjang bagi warga sekitar.

“Pemerintah daerah harus menunjukkan langkah konkret demi menjaga keselamatan dan kesehatan rakyat,” tegasnya lagi.

Isjayanto juga meminta Pemda Gorut untuk tidak berlindung di balik keterbatasan kewenangan dan justru mengabaikan tugas utama dalam melindungi masyarakat.

“Kalau dari sisi kewenangan dianggap kurang memadai, maka temukanlah solusi lain. Jangan hanya pasrah dan menyerah! Keselamatan masyarakat harus menjadi prioritas utama — Salus Populi Suprema Lex Esto,” bebernya.

Ia bahkan menilai, jika hingga kini belum ada tindakan nyata dari pemerintah daerah, patut diduga ada ketidakberpihakan terhadap keselamatan publik.

“Bupati, OPD, hingga Forkopimda harus segera turun langsung ke lapangan. Jangan menunggu jatuhnya korban. Ancaman polusi udara akibat FABA ini nyata dan dampaknya bisa semakin buruk jika dibiarkan terus-menerus,” tutup Isjayanto.

Gorontalo Utara

Angka Fantastis! 1000 Triliun Uang Negara Hilang Setiap Tahun dari Transaksi Gelap Ekspor

Published

4 weeks ago

09/11/2025

Redaksi Barakati

Menguak tabir ekspor Indonesia, terkuak praktik manipulasi transaksi melalui under dan over invoicing yang menyebabkan kerugian negara hingga ribuan triliun rupiah setiap tahun. Dalam podcast Forum Keadilan TV, ekonom sekaligus peneliti Lingkar Studi Perjuangan, Gede Sandra, mengurai bagaimana modus gelap ini terjadi.

Menjelaskan di hadapan host Margi Syarif, Gede Sandra menegaskan dua bentuk utama praktik: “Jadi ini dalam istilah resminya itu namanya misinicing. Misinvoicing iya artinya invoice yang tidak tepatlah kira-kira gitu. Oke. Dan kejadiannya itu ada dua jenis misinicing ini. Yang pertama under invoicing. Oke. Artinya nilainya di bawah dari nilai sebenarnya gitu. Under kan. Dimurah-murahin. Dimurah-murahin. Oke. Yang kedua, over invoicing. Dimahal-mahalin kebalikannya. Oke.”

Modus under invoicing biasanya dipakai untuk mengurangi beban pajak dan kewajiban lainnya dengan cara melaporkan nilai ekspor lebih kecil dibandingkan nilai sesungguhnya di negara tujuan. Sementara over invoicing lekat dengan upaya memindahkan dana secara ilegal keluar negeri lewat transaksi yang nilainya justru dilebihkan. Praktik ini bukan barang baru, telah berlangsung secara konsisten selama 10 tahun terakhir di pemerintahan Presiden Joko Widodo menurut hasil penelitian LSP dan pendataan Next Indonesia.

Pendekatan manipulasi ini kerap menyasar komoditas primadona seperti batu bara, minyak sawit, minyak bumi, hingga logam mulia. Data Dirjen Pajak Kemenkeu terbaru mengungkap temuan 25 wajib pajak pada 2025 menggunakan under invoicing pada ekspor limbah sawit (POME), dengan total transaksi Rp 2,08 triliun. Potensi kerugian negara diperkirakan mencapai Rp 140 miliar hanya dari satu komoditas tersebut.

Lebih jauh, peneliti menyatakan, Dua data ini dibandingkan dan selisihnya inilah yang dianggap ini gelap yang kemudian memunculkan Jadilah angka 1000 triliun. Indikasi klasik under invoicing adalah ketika catatan ekspor Indonesia jauh lebih kecil daripada data impor negara tujuan—hal ini kerap muncul di audit lembaga internasional seperti Global Financial Integrity.

Kekurangan koordinasi antarinstansi seperti Bea Cukai, Kementerian Perdagangan, dan Ditjen Pajak menajamkan celah manipulasi, sehingga pengawasan masih lemah dan potensi kebocoran makin besar. Bimo Wijayanto, Dirjen Pajak, menegaskan, “Kami deteksi di tahun 2025 itu ada sekitar 25 wajib pajak pelaku ekspor yang menggunakan modus yang sama. Ini masih dugaan dari 25 pelaku tersebut setidaknya total transaksinya itu sekitar Rp 2,08 triliun. Jadi, potensi kerugian negara kami estimasi dari Rp 2,08 triliun dari sisi pajak itu sekitar Rp 140 miliar,” ujarnya dalam konferensi pers di Jakarta Utara.

Forum Keadilan TV dan LSP menegaskan urgensi reformasi agar Indonesia tak terus dirugikan melalui praktik manipulasi faktur transaksi lintas negara. Jika tak segera diatasi, kerugian negara bakal semakin menganga, menunda pembenahan ekonomi rakyat.